Counting algebraic tori over ℚ by Artin conductor

In this paper we count the number [Formula: see text] of [Formula: see text]-dimensional algebraic tori over [Formula: see text] whose Artin conductor of the associated character is bounded by [Formula: see text]. This can be understood as a generalization of counting number fields of a given degree by discriminant. We suggest a conjecture on the asymptotics of [Formula: see text] and prove that this conjecture follows from Malle’s conjecture for tori over [Formula: see text]. We also prove that [Formula: see text], and this upper bound can be improved to [Formula: see text] under the assumption of the Cohen–Lenstra heuristics for [Formula: see text].